Professor Ralph Kaufmann
of University of Connecticut, Storrs
will speak on
"Operads, strings topology, and Deligne's conjecture"
In a seminal work Chas and Sullivan introduced operations on the loop space of a compact manifolds which are known as string topology. These operations, which are inspired by the picture of strings moving, breaking up and recombining, are most conveniently described by the operad of cacti defined by Voronov. We will discuss an operadic approach to these operations using an operad based on surfaces with boundary. This description together with theorems relating cacti operads to more classical ones -- the little discs and the framed little discs operad -- allows us to use the same framework to give a solution to Deligne's conjecture on the Hochschild cohomology of an associative algebra, its cyclic generalization to Frobenius algebras, string topology and even give a moduli construction for the renormalization Hopf algebra of Connes and Kreimer. It also allows us to conjecture a wide generalization of string topology to operations of the chains of moduli space on the loop space of a compact manifold. We will begin by introducing two archetypical operads in order to define the notion of operads and to illuminate the ideas behind Deligne's conjecture. Adding extra structures to these basic examples we will discuss the above relations and theorems which link algebra to geometry in a clear and beautiful way.
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